Having trouble understanding minimal polynomial problems

[catsarebad]

i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given.

i'm not exactly sure how you could since you can possibly lose repeated e-values when you write minimal polynomial. how can u create a n dimensional matrix from a polynomial with possibly lower power? idk just a dumb thought.

 

[jgens]

Two distinct matrices (even of different sizes) can have the same minimal polynomial. So there is no "matrix representation" of a minimal polynomial.

 

[HallsofIvy]

Even if all eigenvalues were of "multiplicity" 1, they could correspond to different eigenvectors so to different matrices.

 

[kduna]

i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given.

i'm not exactly sure how you could since you can possibly lose repeated e-values when you write minimal polynomial. how can u create a n dimensional matrix from a polynomial with possibly lower power? idk just a dumb thought.

I wouldn't call it a dumb thought, it is good that you are trying to develop a deeper understanding.

As was mentioned by other posters, distinct matrices can have the same minimal polynomial. In fact, similar matrices have the same minimal polynomial.

Given a minimal polynomial, you can construct particularly nice block matrices which are called rational canonical forms. Now for a given minimal polynomial for some nxn matrix, there is potentially more than one rational canonical forms you could construct. You would need to know more than just the minimal polynomial for your given transformation to determine which rat. canon. form is right for your transformation.

 

[lurflurf]

Let your vector space be polynomials modulo then given polynomial. Let the operator be multiplication by x. The matrix of the operator will have the desires minimal polynomial (of course not unique).